1. Moore Graphs and Beyond: Recent Advances in the Degree/Diameter Problem Mirka Miller King’s College London, UK & University of Newcastle, Australia & University of West Bohemia, Czech Republic & ITB Bandung, Indonesia mirka.miller@gmail.com UPC Barcelona, October 2012 Priority Research Centre for Computer-Assisted Research Mathematics and its Applications Graph Theory and Applications research group

2. 2 Outline of the talk Degree/diameter problem  Some techniques  Some open problems  Prove non-existence of almost Moore digraphs  Construct some missing mixed Moore graphs or prove their non- existence  Construct radial Moore graphs or prove their non-existence

3. 3 Open Problems 1. Does there exist a regular directed graph with degree d>3, diameter k>4 and n=d +d 2 + … + d k vertices? 2. Does there exist a regular mixed graph with undirected degree 3, out-degree 3, diameter 2 and 40 vertices? 3. Find a general construction for undirected graphs of degree 3, order n= (3 k+1 – 1)/2, and diameter at most k+1.  k=0, n=1 k=1, n=4 k=2, n=13  k=3, n=40 k=4, n=121 k=5, n=364 etc.

4. 4 Degree/diameter problem Degree/diameter problem (proposed by Elspas, 1964): Determine the largest number of vertices N(d,k) of a graph G for given maximum degree d and diameter at most k. Directed version: Determine the largest number of vertices N(d,k) of a digraph G for given maximum out-degree d and diameter at most k. Mixed version: Determine the largest number of vertices N(r,z,k) of a mixed graph G for given maximum undirected degree r, maximum out-degree z (d=r+z) and diameter at most k.

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6. 6 Almost Moore Digraphs 1. Do there exist directed graphs with maximum out- degree d, diameter k and the number of vertices one less than the Moore bound M d,k ?

7. 7 Moore bound A natural upper bound on the number of vertices n of digraph G with given maximum out-degree d and diameter at most k is: n  = 1+d +d 2 + … + d k. This bound M d,k is called the Moore bound. A digraph attaining this bound is called a Moore digraph M(d,k).

8. 8 Moore digraphs Examples of Moore digraphs M(d.k) – an undirected edge represents two arcs of opposite directions C3C3 C5C5 Plesnik & Znam, 1974, Bridges & Toueg, 1980 : Moore digraphs exist only in trivial cases, for d =1 (directed cycles of k+1 vertices) or k =1(complete digraphs on d+1 vertices).

9. 9 Almost Moore digraphs Since Moore digraphs do not exist for all d >1 and k >1, we are interested in digraphs that are “close to being Moore”. Relax order to M d,k –  (defect  1  If  = 1 almost Moore digraphs or digraphs of defect 1. Notation. (d,k)-digraph is a digraph of maximum out-degree d, diameter k and n = M d,k –1 vertices. 1 2 56 3 4 1 2 3 4 5 6 1 2 3 4 5 6

10. 10 Almost Moore digraphs Fiol,Alegre,Yebra, ‘83: (d,2)-digraphs exist for any d  2. Example: The line digraph L(K d+1 ) of the complete digraph K d+1 (Kautz digraphs). 56 3 4 1 2 6 2 3 1 5 4 MM,Fris, 1992: There are no (2,k)-digraphs for any k  3. Baskoro,MM,Siran,Sutton, 2005: There are no (3,k)- digraphs for any k  3.

11. 11 Almost Moore digraphs Gimbert in 2001 showed that line digraphs are the only (d,2)-digraphs for d  3. Conde,Gimbert,Gonzalez,Miret,Moreno, 2008: There are no (d,3)-digraphs for any d  3. Do there exist any (d,k)-digraphs, d  4, k  4? Lleida Mafia : NO for k=4. So next smallest unknown case is n=4+16+64+256+1024=1364, d=4, k=5 Now: Do there exist any (d,k)-digraphs, d  4, k  5?

12. 12 Repeats Could be useful for Problems 1 and 2.

13. 13 Almost Moore digraphs: repeats Let G be a (d,k)-digraph.  ( MM&Fris, 1988 ) For every vertex x of G there exists a vertex y, called the repeat of x, such that there are two walks of lengths  k from x to y. Write y = r(x). N+(S): the multiset of all out-neighbours of the vertices in set S. r(S): the multiset of all the repeats of the vertices in S.

14. 14 Neighbourhood Theorem Neighbourhood Theorem. (Baskoro, MM, Plesnik, Znam ’95) Let G be a diregular (d,k)-digraph. Then N + (r(u)) = r(N + (u)). The mapping r is an automorphism, permutation

15. 15 Neighbourhood Theorem Proof. By listing in two different ways the multiset of vertices that can be reached from a vertex v in at most k+1 steps.

16. 16 AMDs: regularity Almost Moore digraphs for any d and k>2 are  out-regular – by simple counting argument  in-regular – not so simple MM, Gimbert, Siran, Slamin, ’00. For k>2, all digraphs of maximum out-degree d>1 and order 1 less than the Moore bound are diregular. So we can use Neighbourhood Theorem for all almost Moore digraphs.

17. 17 Almost Moore digraphs Repeat order of a vertex  =  (v) is the smallest number  such that r  (v) = v. Further study may focus on the existence of  (d,k)-digraphs with selfrepeats  (d,k)-digraphs without selfrepeats

18. 18 Almost Moore digraphs with selfrepeats Theorem. For d>1, k>2, every (d,k)-digraph contains either k selfrepeats or none. [Baskoro, MM & Plesnik, 1998] In other words, there is one C k or none. Note that the girth (length of shortest cycle) of AMD is always k or k+1. Useful lemmas: 1.For any selfrepeat v in a (d,k)-digraph G, the permutation r on N+(v) has the same structure as the permutation r on N  (v). 2.The permutation r on N+(v) has the same structure for every selfrepeat v in a (d,k)-digraph.

19. 19 Almost Moore digraphs with selfrepeats OUTLINE OF THE PROOF. Let V 1 be the set of all selfrepeats in G. Consider the induced subdigraph G[V 1 ]. Since in G any walk of length  k joining any two selfrepeat vertices involves only selfrepeats then the diameter G[V 1 ] is at most k. If all vertices of G[V 1 ] have outdegree 1 then there are exactly k vertices in G[V 1 ]; and G[V 1 ] is a k-cycle. If there exists a vertex in G[V 1 ] with outdegree d 1  2 then G[V 1 ] is diregular, degree d 1 and order d 1 +d 1 2 +…+d 1 k almost Moore digraph! BUT…

20. 20 Almost Moore digraphs with selfrepeats OUTLINE OF THE PROOF that there are no (d 1,k)- digraphs with all vertices selfrepeats, d 1 >1, k>2. I+A+…+A k =J+I, where A is the adjacency matrix of G, J is unit matrix, I is identity matrix A+…+A k =J J has eigenvalues n (once) and 0 (n-1 times) A has eigenvalues d (once) and n-1 roots of the characteristic equation x+x 2 +… +x k = 0 x = 0 and roots of x k -1 =0 Since tr(A j )=0 for 0  j  k-1, we obtain –d = –d k-1 and so d =1 or k =2 are the only solutions.

21. 21 Almost Moore digraphs d  4, k  5  (d,k)-digraphs with selfrepeats There are k selfrepeats Given the repeats structure of the out-neighbourhood of one selfrepeat vertex, we can determine the repeats structure of the digraph… this could be helpful…  (d,k)-digraphs without selfrepeats Totally open Current directions to prove non-existence: Lleida using algebraic methods; Pilsen divisibility conditions based on the lengths of small cycles; Malang based on vertex types and counting in two different ways – none finished.

22. 22 Mixed Moore Graphs 2. Does there exist a regular mixed graph with degree 3, out-degree 3, diameter 2 and 40 vertices?

23. 23 Moore bound - directed Let G be a digraph with given out-degree d and (a) diameter at most k (b) girth at least k+1. What is the (a) maximum (b) minimum possible number of vertices in G? n  M* d,k = 1+d +d 2 + … + d k v This bound is called the Moore bound (for digraphs). A digraph attaining this bound is called a Moore digraph. Plesnik & Znam, ’74, Bridges & Toueg, ’80 : Moore digraphs exist only for trivial cases, for d =1 (directed cycles of k+1 vertices) or k =1 (complete digraphs on d+1 vertices). No open problems.

24. 24 Moore bound Let G be a graph of degree d and (a) diameter at most k (b) girth at least 2k + 1. What is the (a) maximum (b) minimum number of vertices in G? n  M d,k =1+d+d(d-1)+…+d(d-1) k-1 This bound is called the Moore bound.. Graph attaining this bound is called a Moore graph. v

25. 25 Moore graphs k =1: Moore graphs are complete graphs K d+1. k =2: Moore graphs exist for d =2 (cycle C 5 ) or d =3 (Petersen graph) or d =7 (Hoffman-Singleton graph) or d =57(?) k =3: Moore graph exists for d =2 (cycle C 7 ). Hoffman&Singleton,’60 k  3: Moore graphs are C 2k+1. Damerell, ‘73; Bannai & Ito, ’73 One open problem: k=2, d=57.

26. 26  A mixed graph may contain (undirected) edges as well as (directed) arcs.  A proper mixed graph has at least 1 edge and 1 arc. Mixed graphs v E3E3 E2E2 E4E4 E1E1 E5E5

27. 27 Moore bound for mixed graphs v If r=0 then d=z and get directed Moore bound If z=0 d=r undirected Moore bound Theorem. There are no mixed Moore graphs of diameter > 2. Nguyen, Gimbert, MM, ‘07

28. 28 Mixed Moore graphs of diameter 2 Bosak graph: Proper mixed Moore graph of order 18, z=1, r=3. Kautz digraph: Proper mixed Moore graph of order 12, z=2, r=1.

29. 29 Mixed Moore graphs of diameter 2 (Bosák,1979) A mixed Moore graph of order n, degree d and diameter k=2 has the following properties: i. Totally regular, undirected degree r and directed degree z (d=z+r) ii. iii. either or or Many open problems: k=2, many d.

30. 30 Mixed Moore graph s Mixed Moore graph with n=40, d=6, z=3, r=3, k=2 can be considered as a regular digraph of diameter 2, degree 6, and M 6,2 – 3 vertices, with every vertex having 3 selfrepeats. Recent new result: Jorgensen constructed mixed Moore graph of 108 vertices with undirected degree 3, directed degree 7 and diameter 2.

31. 31 Mixed Moore graphs of diameter 2 ndzrExampleUniqueness (up-to-isomorphism) 3110Yes 5202C5C5 6211Ka(2)Yes 10303Petersen graphYes 12321Ka(3)Yes 18413Bosak graphYes 20431Ka(4)Yes 30541Ka(5)Yes 40633?? 42651Ka(6)Yes 50707Hoffman-Singleton graphYes 54743?? 56761Ka(7)Yes 72871Ka(8)Yes 84927?? 88963?? 90981Ka(9)Yes

32. 32 Mixed Moore graphs of diameter 2 (cont.) ndzrExampleUniqueness (up-to-isomorphism) 1041037?? 1061055?? 1081073Jorgensen graph? 1101091Ka(10)Yes 1261147?? 1281165?? 1301183?? 13211101Ka(11)Yes 1501257?? 1521275?? 1541293?? 15612111Ka(12)Yes 1761367?? 1781385?? 18013103?? 18213121Ka(13)Yes 19814113??

33. 33 Mixed Moore graphs of diameter 2 (cont.) ndzrExampleUniqueness (up-to-isomorphism) 2041477?? 2061495?? 20814113?? 21014131Ka(14)Yes 22815213?? 2341587?? 23615105?? 23815123?? 24015141Ka(15)Yes 26016313?? 2661697?? 26816115?? 27016133?? 27216151Ka(16)Yes 29417413?? 30017107??

34. 34 Radial Moore Graphs 3. Find a general construction for undirected graphs of degree 3, order n= (3 k+1 – 1)/2, and diameter k+1.

35. 35 Radial Moore graphs and digraphs Digraph G is a radially Moore digraph if it has maximum out-degree d, radius k, order M d,k and the diameter of G does not exceed k+1. (Knor,1996) Radially Moore digraph of degree d and radius k exists for every d and k. What about (undirected) radial Moore graphs? G is a radial Moore graph if it has degree d, radius k, order M d,k and the diameter of G does not exceed k+1.

36. 36 Radial Moore graphs (Knor,1996) Radial Moore digraph of degree d and radius k exists for every d and k. What about radial Moore graphs? G is a radial Moore graph if it has degree d, radius k, order M d,k = 1+d+d(d-1)+…+d(d-1) k-1 and the diameter of G does not exceed k+1. (Exoo, Gimbert, Lopez, Gomez, 2012) Radial Moore graph of degree d and radius k=3 exists for every d. (with some input from Knor) Current general conjecture due to Exoo: Apart from the known radial Moore graphs no others exist. Anti-Exoo conjecture: There are more radial Moore graphs.

37. 37 Radial Moore graphs All cubic radial Moore graphs of radius 2.

38. 38 Radial Moore graphs What about cubic radial Moore graphs of radius greater than 2? Cases k=3, 4, 5 are known (Exoo, Baladram) What about k>5? Do they all exist? Why do we care?

39. 39 Current table of largest known graphs

40. 40 Radial Moore graphs Why do we care about cubic radial Moore graphs? We would get infinitely many new largest graphs current largestradial Moore k=8 336 382 k=9 600 766 k=1012501534 …

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